chapter07 analysis of risk and return
TRANSCRIPT
CONTEMPORARY FINANCIAL MANAGEMENT
Chapter 7:
Analysis of Risk and Return
INTRODUCTION
This chapter develops the risk-return relationship for both individual projects (investments) and portfolios of projects
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RISK AND RETURN
Risk is usually defined as the actual or potential variability of returns from a project or portfolio
Risk-free returns are known with certainty Federal Government Treasury Bills are often considered the
risk-free security. The risk-free rate of return sets a floor under all other returns
in the market.
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HOLDING PERIOD RETURN
Return for holding an investment for one period (i.e. period of days, months, years, etc.)
When there is no cash flow during the holding period, then:
4
1 0
0
P - P HPR =
P
HPR = Holding Period ReturnP1 = Ending PriceP0 = Beginning Price
HOLDING PERIOD RETURN When there is a cash flow in addition to the ending price
(such as the payment of a dividend), the Holding Period Return formula is:
( )1 0
0
P +Cash Flow - P HPR =
P
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HOLDING PERIOD RETURN: EXAMPLE You bought a stock one year ago for $10. Today, it is worth $12.
Yesterday, you received a $1 dividend. What is your holding period return?
( )
( )+ −=
=
1 0
0
P +Cash Flow - P HPR =
P
12 1 10
100.30 or 30%
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RETURNS
Ex Post Returns (After the fact) Return that an investor actually realizes
Ex Ante Returns (Before the fact) Return that an investor expects to earn
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ANALYZING RETURN
Expected Return ( ): When returns are not known with certainty, there will often
exist a probability distribution of possible returns with an associated probability of occurrence.
Expected return is a weighted average of the individual possible returns (rj), with weights being the probability of occurrence (pj).
8=
= ∑n
j jj 1
r̂ rp
r̂
EXPECTED RETURN: EXAMPLE
Possible Return
Probability of Occurrence
-10% 5%
0% 10%
+5% 25%
+15% 50%
+25% 10%
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EXPECTED RETURN: SOLUTION
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
=
=
= − + +
+ +
=
∑n
j jj 1
r̂ rp
10% .05 0% .10
5% .25 15% .50 25% .10
10.75%
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ANALYZING RISK Standard Deviation ( ): a statistical measure of the
dispersion, or variability, of outcomes around the mean or expected value ( ).
Low standard deviation means that returns are tightly clustered around the mean
High standard deviation means that returns are widely dispersed around the mean
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σ
r̂
CALCULATING STANDARD DEVIATION
Three common ways of calculating standard deviation:
Returns are known with certainty Standard deviation of a population Standard deviation of a sample
Returns are not known with certainty
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STANDARD DEVIATION OF A POPULATION
( ) ( ) ( )2 2 2
1 2 N2r - r + r - r +...+ r - r
σ =N
2σ = σ
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The standard deviation ( ) is the square root of the variance:
σ
First, calculate the variance ( 2):σ
i
r = Mean returnr = Return i
N = Number of returns
STANDARD DEVIATION OF A SAMPLE
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( ) ( ) ( )2 2 2
1 2 N2r - r + r - r +...+ r - r
S = N - 1
2s = S
First, calculate the variance (S2):
The standard deviation (s) is the square root of the variance:
STANDARD DEVIATION: EXAMPLE
You have been given the following sample of stock returns, for which you would like to calculate the standard deviation:
{12%, -4%, 0%, 22%, 5%}
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STANDARD DEVIATION: EXAMPLE
You have been given the following sample of stock returns, for which you would like to calculate the standard deviation:
{12%, -4%, 0%, 22%, 5%}
Step 1: Calculate Arithmetic Return
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+ + +=
=
1 2 Nr + r + ... + rAM =
N12 - 4 0 22 5
57%
STANDARD DEVIATION: EXAMPLE
You have been given the following sample of stock returns, for which you would like to calculate the standard deviation:
{12%, -4%, 0%, 22%, 5%}
Step 2: Calculate Variance
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( ) ( ) ( )
( ) ( ) ( ) ( ) ( )− + − − + − + − + −=
−=
2 2 2
1 2 N2
2 2 2 2 2
r - r + r - r +...+ r - rS =
N-1
12 7 4 7 0 7 22 7 5 7
5 1106
STANDARD DEVIATION: EXAMPLE
You have been given the following sample of stock returns, for which you would like to calculate the standard deviation:
{12%, -4%, 0%, 22%, 5%}
Step 3: Calculate Standard Deviation
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==
2s = S
10610.3%
STANDARD DEVIATION – RETURNS NOT CERTAIN
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( )∑ $n 2
2j j
j=1
σ = r - r p
rj = return at time period j
= expected return
pj = probability of return j occurring
r̂
STANDARD DEVIATION: EXAMPLE You have been provided with the following possible returns
and their associated probabilities. Calculate the expected return and the standard deviation of return.
State of Economy Return Probability
Boom 30% 15%
Normal 15% 60%
Recession 0% 25%
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STANDARD DEVIATION: SOLUTION
Step #1: Calculate the Expected Return
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( ) ( ) ( ) ( ) ( ) ( )=
=
= + +
=
∑n
j jj 1
r̂ rp
30% .15 15% .60 0% .25
13.5%
STANDARD DEVIATION: SOLUTION
Step #2: Calculate the Variance
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( )( ) ( ) ( ) ( ) ( ) ( )
∑ $n 2
2j j
j=1
2 2 2
σ = r - r p
= 30-13.5 .15 + 15-13.5 .60 + 0-13.5 .25
= 87.75
STANDARD DEVIATION: SOLUTION
Step #3: Calculate the Standard Deviation
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2σ = σ
= 87.75= 9.4%
NORMAL PROBABILITY DISTRIBUTION
A symmetrical, bell-like curve where 50% of possible outcomes are greater than the expected value and 50% are less than the expected value.
A normal distribution is fully described by just two statistics: Mean Standard deviation
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NORMAL PROBABILITY DISTRIBUTION
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Mean- 1 σ
50% Probability
- 3 σ - 2 σ 1 σ 2 σ 3 σ
50% Probability
68.26%
95.44%
99.74%
NORMAL DISTRIBUTION EXAMPLE
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10%
50% Probability
5% 15%0% 20%- 5% 25%
50% Probability
68.26%
95.44%
99.74%
Mean = 10%; Standard Deviation = 5%
STANDARD NORMAL PROBABILITY
Problem: Standard deviation is correlated with size of the mean
Solution: To allow for easy comparison among distributions with different means, standardize using a Z score
Z score measures the number of standard deviations ( ) a particular rate of return (r) is from the mean or expected value ( ).
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−=σ
ˆr rz
r̂σ
STANDARD NORMAL: EXAMPLE What is the probability of a loss on an investment with an
expected return of 20% and a standard deviation of 17%?
Step #1: Calculate the Z Score for the number of standard deviations from the mean for a 0% return
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− −= = ≅ −σ
ˆr r 0% 20%z 1.18
17%
STANDARD NORMAL: EXAMPLE What is the probability of a loss on an investment with an expected
return of 20% and a standard deviation of 17%?
Step #2: Consult Table V on Page 712. Find the row with 1.10 in the left hand column. Then find the column with 0.08 in the top row. The cell where the row & column intersect is the probability of obtaining a value less than 1.18 standard deviations from the mean. The answer is 0.1190 or 11.90%
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PROBABILITY OF EARNING LESS THEN 0%
30
20%3%- 31% -14% 37% 54% 71%
1.18 st. dev.from the mean
11.9% Probability
CONCEPT OF EFFICIENT PORTFOLIOS
Has the highest possible expected return for a given level of risk (or standard deviation)
Has the lowest possible level of risk for a given expected return
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σ
r̂
A B
C A dominates B because it has the same expected return for a given risk.
C dominates B because it has a higher expected return for a given risk.
COEFFICIENT OF VARIATION (V)
The ratio of the standard deviation ( ) to the expected value ( ).
Tells us the risk per unit of return.
An appropriate measure of total risk when comparing two investment projects of different size.
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σ=vr̂
σr̂
COEFFICIENT OF VARIATION: EXAMPLE You are asked to rank the following set of investments
according to their risk per unit of return.
Security Return Standard Deviation
A 6% 7%
B 10% 13%
C 18% 21%
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COEFFICIENT OF VARIATION: SOLUTION
Security Return Standard Deviation
Coefficient of Variation
A 6% 7%
B 10% 13%
C 18% 20%
=71.17
6
=201.1
18=13
1.310
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Most Risk
Least Risk
RELATIONSHIP BETWEEN RISK AND RETURN
Required Rate of Return Discount rate used to present value a stream of expected cash
flows from an asset.
Risk-Return Relationship The riskier, or the more variable, the expected cash flow
stream, the higher the required rate of return.
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RELATIONSHIP BETWEEN RISK AND RETURN
Required Rate of Return =
Risk-free Rate of Return + Risk Premium
Risk-free Rate: rate of return on securities that are free of default risk, such as T-bills.
Risk Premium: expected “reward” the investor expects to earn for assuming risk
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RISK-FREE RATE OF RETURN
Risk-free Rate of Return (rf) =
Real Rate of Return + Exp. Inflation Premium
Real Rate of Return: the reward for deferring consumption
Expected Inflation Premium: compensates investors for the loss of purchasing power due to inflation
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TYPES OF RISK PREMIUMS
Maturity risk premium
Default risk premium
Seniority risk premium
Marketability risk premium
Business risk
Financial risk
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TERM STRUCTURE OF INTEREST RATES Term structure is a plot of the yield on securities with similar
risk but different maturities
Term structure is used to explain the maturity risk premium (why long securities tend to have higher yields than short maturity securities)
Three theories of the Term Structure: Expectations theory Liquidity premium theory Market segmentation theory
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CANADA’S TERM STRUCTURE
40
2.65%
2.69% 2.76%
2.70% 2.91%
3.28%3.57%
4.13% 4.88%
5.39%
0%
1%
2%
3%
4%
5%
6%
1Month
2Month
3Month
6Month
12Months
2 year
3 year
5 year
10 year
Longterm
November 13, 2003
EXPECTATIONS THEORY
The long interest rate is the geometric average of expected future short interest rates.
If the term structure is sloping up, future short interest rates are expected to be higher than current short interest rates.
If the term structure is sloping down, future short interest rates are expected to be lower than current short interest rates.
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LIQUIDITY PREMIUM THEORY
Investors prefer liquidity (the ability to convert to cash at or near face value)
Long securities are less liquid than short securities
Therefore, to induce investors to hold long securities, must pay a “liquidity premium”
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MARKET SEGMENTATION THEORY
The yield in each segment of the yield curve is determined by the supply and demand for funds in that maturity zone
Supply & demand driven by firms which deal primarily in a specific maturity zone Chartered banks – short maturities Trust companies – medium maturities Pension funds – long maturities
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MODERN PORTFOLIO THEORY
44Harry Markowitz William F. Sharpe Merton Miller
Modern portfolio theory was introduced by Harry Markowitz in 1952. Markowitz, Sharpe & Miller were co-recipients of the Nobel Prize in Economics in 1990 for their pioneering work in portfolio theory
EXPECTED RETURN The expected return on a portfolio is the weighted average of
the returns of each asset within the portfolio
Example: A portfolio is comprised of three securities with the following returns:
Security Return % of Portfolio
A 5% 30%
B 10% 45%
C 15% 25%
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EXPECTED RETURN
The expected return of the portfolio is the weighted average:
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( ) ( ) ( ) ( ) ( ) ( )=
=
= + +
=
∑n
j jj 1
r̂ rw
5% .30 10% .45 15% .25
9.75%
rj = return at time period j
= expected return
wj = proportion of the portfolio comprised of asset j
r̂
PORTFOLIO RISK: TWO RISKY ASSETS
Standard deviation of a two-asset portfolio is calculated as follows:
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2 2 2 2A A B B A B A,B A Bσ = w σ + w σ + 2w w ρ σ σ
A
A,B
σ = standard deviationw = the proportion of the portfolio comprised of A
ρ = the correlation coefficient between A & B
PORTFOLIO RISK: TWO RISKY ASSETS
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2 2 2 2A A B B A B A,B A Bσ = w σ + w σ + 2w w ρ σ σ
Portfolio risk is driven mainly by the correlation between the
assets!!
CORRELATION
Correlation is a measure of the linear relationship between two assets
Correlation varies between perfect negative (-1) to perfect positive (+1)
Perfect negative correlation: when the return on asset A rises, the return on Asset B falls and vice versa
Perfect positive correlation: the returns on asset A and Asset B move in perfect unison
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CORRELATION & RISK REDUCTION
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B
ARA
σA σB
RBPerfect Negative Correlation
Perfect Positive Correlation
Less thanPerfect Correlation
To minimize portfolio risk, choose assets that have very low correlations with each other.
MOVING TOWARD MANY RISKY ASSETS
When the portfolio consists of many risky assets, they form a plot similar to a broken egg shell shape
Each dot within the broken egg shell shape represents the risk/return profile for a single risky asset or portfolio of risky assets
To maximize return per unit of risk assumed, an investor would always choose an asset or portfolio that plots along the efficient frontier
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PORTFOLIOS: MANY RISKY ASSETS
52
You would never choose Asset A, as you can earn a higher return with similar risk by choosing the asset that plots along the Efficient Frontier.
RA
σA
Return
Standard Deviation
A
CHOOSING A PORTFOLIO: SO FAR
The investor first decides how much risk to assume
The investor then chooses the portfolio that plots along the efficient frontier with that amount of risk
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INTRODUCING THE RISK FREE SECURITY
When a risk-free asset (Treasury Bill) is introduced into the set of risky assets, a new efficient frontier emerges
This new efficient frontier is known as the Capital Market Line (CML)
The CML represents all possible portfolios comprised of Treasury Bills and the Market Portfolio
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ADDING THE RISK-FREE ASSET
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RM
σM
Return
Standard Deviation
ARf
Capital Market Line
CAPITAL MARKET LINE (CML)
To maximize return for an amount of risk, investors should hold a portion of their assets in T-bills and a portion in the market portfolio.
Linear relationship between risk and return
To earn an expected return greater than the return on the market portfolio, invest more than 100% of one’s own wealth in the market portfolio.
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( )PortfolioWell-diversified f Market fPortfolio Market
σR = r + r - r
σ
WHAT ARE WE MISSING? We know:
Investors should split their assets between Treasury bills and the market portfolio
To reduce risk, invest a greater proportion of assets in Treasury bills
To enhance expected return, invest a greater proportion of assets in the market portfolio
We do not know how to calculate the expected return (and hence the price) for a single risky asset The Capital Asset Pricing Model is needed
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THE MISSING LINK
We need to measure the Market risk that cannot be diversified away
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Unique or Non-systematic
Risk
Market or Systematic
Risk
Diversifiable Non- Diversifiable
Total Standard Deviation (or Risk)
THE MISSING LINK
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Unique Risk Market Risk(measured with Beta)
The market will not compensate us for risk that
can be diversified away.
The market will compensate us for market risk – the risk that
cannot be diversified away
CAPM: SYSTEMATIC RISK IS RELEVANT
Systematic, or non-diversifiable, risk is caused by factors affecting the entire market interest rate changes changes in purchasing power change in business outlook
Unsystematic, or diversifiable, risk is caused by factors unique to the firm strikes regulations management’s capabilities
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PORTFOLIO DIVERSIFICATION When assets are put into a well-diversified portfolio, some of
the unique or nonsystematic risk is diversified away
The number of assets required to diversify away most of the unique risk varies with the correlation between the assets Canada’s capital markets are more highly correlated with the
natural resource sector than are the US capital markets Require more securities in Canada to diversify away most of the
unique risk
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DIVERSIFYING UNIQUE RISK
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Risk
Unique Risk
Number of Securities
Portfolio Risk
Market Risk
SYSTEMATIC RISK IS MEASURED BY BETA
Beta is a measure of the volatility of a security’s return compared to the volatility of the return on the Market Portfolio
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= Covariancej,MarketSecurity j VarianceMarket
β
CONCEPT OF BETA
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To calculate Beta, use historical monthly
rates of return on both the security and the
market index.
Return on Stock A
Return on TSX Index
Slope equals Beta
SECURITY MARKET LINE (SML)
Shows the relationship between required rate of return and beta (ß).
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rf
ß
Security Market Line
RequiredRate ofReturn
ßj
k j
REQUIRED RATE OF RETURN The required return for any security j may be defined in terms
of systematic risk, βj, the expected market return, rm, and the expected risk free rate, rf.
66
= + −j jf m fkβ ( )ˆ ˆ ˆr r r
^
^
SML: EXAMPLE
A security has a Beta of 1.25. If the yield on Treasury Bills is 5% and the return on the market portfolio is 11%, what is the expected return for holding the security?
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An investor expects a return of 12.5% to hold the security.
MARKET RISK PREMIUM
The reward for bearing risk Equal to (rm – rf)
Equal to the slope of security market line (SML)
Will increase or decrease with uncertainties about the future economic outlook the degree of risk aversion of investors
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SECURITY MARKET LINE (AGAIN)
69β
SMLReturn
βM
RM
A
B
A – return is too high; price is too low
B – return is too low; price is too high
CAPM ASSUMPTIONS
Investors hold well-diversified portfolios
Competitive markets
Borrow and lend at the risk-free rate
Investors are risk averse
No taxes
Investors are influenced by systematic risk
Freely available information
Investors have homogeneous expectations
No brokerage charges 70
CAPM DRAWBACKS
Estimating expected future market returns on historic returns.
Determining an appropriate rf
Determining the best estimate of β
Investors don’t totally ignore unsystematic risk
Betas are frequently unstable over time
Required returns are determined by macroeconomic factors
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MARKET EFFICIENCY
Capital markets are efficient if prices adjust fully and instantaneously to new information affecting a security’s prospective return.
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THREE DEGREES OF MARKET EFFICIENCY
Weak form
Semi-strong form
Strong form
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WEAK FORM MARKET EFFICIENCY
Security prices capture all of the information contained in the record of past prices and volumes
Implication: No investor can earn excess returns using historical price or volume information. Technical analysis should have no marginal value.
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SEMI-STRONG FORM MARKET EFFICIENCY
Security prices capture all of the information contained in the public domain.
Implication: No investor can earn excess returns using publicly available information. Fundamental analysis should have no marginal value.
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STRONG FORM MARKET EFFICIENCY
Security prices capture all information, both public and private.
Markets are quite efficient (but it is illegal to use private information for personal gain, when trading securities)!
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